cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A193778 Number of signed permutations of length 2n invariant under D and D'bar.

Original entry on oeis.org

1, 2, 8, 32, 160, 832, 4864, 29696, 195584, 1341440, 9723904, 73105408, 574062592, 4657184768, 39165624320, 339133595648, 3028204650496, 27760959422464, 261439835078656, 2521668748574720, 24912764963127296, 251559029812232192, 2595790316527157248
Offset: 0

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Author

Andy Hardt, Aug 04 2011

Keywords

Comments

See section 6 in the Hardt reference.

Examples

			For n = 1, the permutations of length 2 are (-1, 2) and (1, -2).

Links

Crossrefs

Column k=4 of A376826.
Cf. A193777, A000085.

Programs

  • Maple
    exp(2*x*(x + 1)): series(%, x, 23):
seq(n!*coeff(%, x, n), n = 0..22); # Peter Luschny, May 30 2021
  • Mathematica
    RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2a[n-1]+4(n-1)a[n-2]},a,{n,30}] (* Harvey P. Dale, Mar 14 2021 *)
a[n_] := 2^n HypergeometricPFQ[{-n/2, (1-n)/2}, {}, 2];
Table[a[n], {n, 0, 22}] (* Peter Luschny, May 30 2021 *)
  • PARI
    seq(n)={my(v=vector(n+1)); v[1]=1; v[2]=2; for(n=2, n, v[n+1] = 2*v[n] + 4*(n-1)*v[n-1]); v} \\ Andrew Howroyd, Dec 08 2018

Formula

a(n) = 2*a(n-1) + 4*(n - 1)*a(n-2).
From Peter Luschny, May 30 2021: (Start)
a(n) = 2^n * hypergeom([-n/2, (1 - n)/2], [], 2).
a(n) = 2^n * A000085(n).
a(n) = n! * [x^n] exp(2*x*(x + 1)). (End)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 08 2018

A300700 Triangle read by rows: T(n, n-k) = number of k-faces of the concertina n-cube.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 18, 42, 26, 1, 58, 252, 344, 150, 1, 190, 1420, 3380, 3230, 1082, 1, 614, 7770, 29200, 47130, 34452, 9366
Offset: 0

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Author

Tilman Piesk, Mar 11 2018

Keywords

Comments

n-place formulas in first-order logic like Ax Ey P(x, y) can be ordered by implication. This Hasse diagram can be interpreted as an n-dimensional convex polytope with face dimensions ranging from 0 (the vertices) to n (the polytope itself).
The right diagonal (n-k = 0, number of vertices) is A000629, which is twice an ordered Bell number (A000670) for n>0.
The second right diagonal (n-k = 1, number of edges) is A300693.
The second left diagonal (k = 1, number of facets) is 2, 6, 18, 58, 190, 614, ... (not to be confused with A151282 or A193777).
The third left diagonal (k = 2, number of ridges) is 6, 42, 252, 1420, 7770, ...
The row sums are A300701. The central diagonal starts 1, 6, 252, 29200 and the row maxima start 1, 2, 6, 42, 344, 3380, 47130.
The corresponding triangle for hypercubes is A013609, and its row sums are A000244 (powers of 3). That for permutohedra is A019538, and its row sums are A000670 (ordered Bell numbers).

Examples

			First rows of the triangle:
  k      0     1     2     3     4     5     6         sums = A300701
n
0        1                                                1
1        1     2                                          3
2        1     6     6                                   13
3        1    18    42    26                             87
4        1    58   252   344   150                      805
5        1   190  1420  3380  3230  1082               9303
6        1   614  7770 29200 47130 34452  9366       128533
T(3, 3-1) = T(3, 2) = 42 is the number of 1-faces (edges) of the concertina 3-cube. It has 26 vertices, 42 edges, 18 faces and 1 cell.
In the reflected triangle the column number is the dimension of the counted faces:
  n-k    0     1     2     3     4     5     6
n
0        1
1        2     1
2        6     6     1
3       26    42    18     1
4      150   344   252    58     1
5     1082  3230  3380  1420   190     1
6     9366 34452 47130 29200  7770   614     1

Links

Crossrefs

Cf. A300701, A000629, A300693.
Cf. A013609, A000244 (for hypercubes).
Cf. A019538, A000670 (for permutohedra).

A321629 The Riordan square of the permutation involutions. Triangle T(n, k), 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 4, 8, 5, 1, 10, 22, 18, 7, 1, 26, 62, 60, 32, 9, 1, 76, 184, 194, 126, 50, 11, 1, 232, 568, 630, 462, 228, 72, 13, 1, 764, 1840, 2072, 1644, 938, 374, 98, 15, 1, 2620, 6204, 6972, 5788, 3650, 1710, 572, 128, 17, 1
Offset: 0

Views

Author

Peter Luschny, Dec 09 2018

Keywords

Comments

The Riordan square is defined in A321620.

Examples

			[0]    1;
[1]    1,    1;
[2]    2,    3,    1;
[3]    4,    8,    5,    1;
[4]   10,   22,   18,    7,    1;
[5]   26,   62,   60,   32,    9,    1;
[6]   76,  184,  194,  126,   50,   11,   1;
[7]  232,  568,  630,  462,  228,   72,  13,   1;
[8]  764, 1840, 2072, 1644,  938,  374,  98,  15,  1;
[9] 2620, 6204, 6972, 5788, 3650, 1710, 572, 128, 17, 1;

Crossrefs

First column are the self-inverse permutations A000085.
Row sums are A193777, alternating row sums are A000007.
Cf. A321620.

Programs

  • Maple
    # The function RiordanSquare is defined in A321620.
cf := proc(dim) local k, m; m := 1;
for k from dim by -1 to 1 do m := 1 - k*x - k*x^2/m od;
1/m end: RiordanSquare(cf(9), 9);
Showing 1-3 of 3 results.

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